Reidemeister Torsion for Vector Bundles on subscriptsuperscriptℙ1ℤ
We consider vector bundles of rank 2 with trivial generic fiber on the projective line over ℤ . For such bundles, a new invariant is constructed — the Reidemeister torsion, which is an analog of the classical Reidemeister torsion from topology. For vector bundles of rank 2 with trivial generic fiber...
Saved in:
| Published in | Proceedings of the Steklov Institute of Mathematics Vol. 325; no. Suppl 1; pp. S155 - S167 |
|---|---|
| Main Author | |
| Format | Journal Article Conference Proceeding |
| Language | English |
| Published |
Moscow
Pleiades Publishing
01.08.2024
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0081-5438 1531-8605 |
| DOI | 10.1134/S008154382403012X |
Cover
| Abstract | We consider vector bundles of rank
2
with trivial generic fiber on the projective line over
ℤ
. For such bundles, a new invariant is constructed — the Reidemeister torsion, which is an analog of the classical Reidemeister torsion from topology. For vector bundles of rank 2 with trivial generic fiber and jumps of height 1, that is, for the bundles that are isomorphic to
superscript
𝒪
2
in the fiber over
ℚ
and are isomorphic to
superscript
𝒪
2
or
direct-sum
𝒪
1
𝒪
1
over each closed point of Spec
ℤ
, we calculate this invariant and show that it, together with the discriminant of the bundle, completely determines such a bundle. |
|---|---|
| AbstractList | We consider vector bundles of rank 2 with trivial generic fiber on the projective line over ℤ. For such bundles, a new invariant is constructed — the Reidemeister torsion, which is an analog of the classical Reidemeister torsion from topology. For vector bundles of rank 2 with trivial generic fiber and jumps of height 1, that is, for the bundles that are isomorphic to superscriptð'ª2 in the fiber over ℚ and are isomorphic to superscriptð'ª2 or direct-sumð'ª1ð'ª1 over each closed point of Specℤ, we calculate this invariant and show that it, together with the discriminant of the bundle, completely determines such a bundle. We consider vector bundles of rank 2 with trivial generic fiber on the projective line over ℤ . For such bundles, a new invariant is constructed — the Reidemeister torsion, which is an analog of the classical Reidemeister torsion from topology. For vector bundles of rank 2 with trivial generic fiber and jumps of height 1, that is, for the bundles that are isomorphic to superscript 𝒪 2 in the fiber over ℚ and are isomorphic to superscript 𝒪 2 or direct-sum 𝒪 1 𝒪 1 over each closed point of Spec ℤ , we calculate this invariant and show that it, together with the discriminant of the bundle, completely determines such a bundle. |
| Author | Polyakov, V. M. |
| Author_xml | – sequence: 1 givenname: V. M. surname: Polyakov fullname: Polyakov, V. M. email: polyakov@pdmi.ras.ru organization: St. Petersburg Department of the Steklov Mathematical Institute of the Russian Academy of Sciences |
| BookMark | eNplkM1Kw0AUhQepYFt9AHcB19F75yfJLLXYKhQErdJdSDJ3JKUmcSZ5g76Ee5-sT2JCBBeuzuWcj3vgzNikqiti7BLhGlHImxeABJUUCZcgAPn2hE1RCQyTCNSETYc4HPIzNvN-ByBVLPWULZ-pNPRBpW_JBZva-bKuAlu74I2Ktpe7rjJ78kHv-i73hSub1ncNufE8Hr7wePg-Z6c223u6-NU5e13ebxYP4fpp9bi4XYcNStyGqsgExkJZbSXIzHCyPDaU6UITilhLUDZXhS0oz02Sixgt14iRJRMpbrSYs6vxb-Pqz458m-7qzlV9ZSpAK1QgOO8pPlK-cWX1Tu6PQkiHvdJ_e4kf0-th1A |
| ContentType | Journal Article Conference Proceeding |
| Copyright | Pleiades Publishing, Ltd. 2024 Pleiades Publishing, Ltd. 2024. |
| Copyright_xml | – notice: Pleiades Publishing, Ltd. 2024 – notice: Pleiades Publishing, Ltd. 2024. |
| DOI | 10.1134/S008154382403012X |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics |
| EISSN | 1531-8605 |
| EndPage | S167 |
| ExternalDocumentID | 10_1134_S008154382403012X |
| GroupedDBID | -5D -5G -BR -EM -Y2 -~C -~X .VR 06D 0R~ 0VY 123 1N0 29P 2J2 2JN 2JY 2KG 2KM 2LR 2VQ 2~H 30V 4.4 408 40D 40E 5VS 6NX 78A 8TC 8UJ 95- 95. 95~ 96X AAAVM AABHQ AACDK AAHNG AAIAL AAJBT AAJKR AANZL AARHV AARTL AASML AATNV AATVU AAUYE AAWCG AAYIU AAYQN AAYTO AAYZH ABAKF ABBBX ABDZT ABECU ABFTV ABHLI ABHQN ABJNI ABJOX ABKCH ABKTR ABMNI ABMQK ABNWP ABQBU ABQSL ABSXP ABTEG ABTHY ABTKH ABTMW ABULA ABWNU ABXPI ACAOD ACBXY ACDTI ACGFO ACGFS ACHSB ACHXU ACIWK ACKNC ACMDZ ACMLO ACOKC ACOMO ACPIV ACREN ACZOJ ADHHG ADHIR ADINQ ADKNI ADKPE ADRFC ADTPH ADURQ ADYFF ADYOE ADZKW AEBTG AEFQL AEGAL AEGNC AEJHL AEJRE AEKMD AEMSY AENEX AEOHA AEPYU AETLH AEVLU AEXYK AFBBN AFFNX AFGCZ AFLOW AFQWF AFWTZ AFYQB AFZKB AGAYW AGDGC AGJBK AGMZJ AGQMX AGRTI AGWIL AGWZB AGYKE AHAVH AHBYD AHKAY AHSBF AHYZX AIAKS AIGIU AIIXL AILAN AITGF AJBLW AJRNO ALMA_UNASSIGNED_HOLDINGS ALWAN AMKLP AMTXH AMXSW AMYLF AMYQR AOCGG ARMRJ ASPBG AVWKF AXYYD AZFZN B-. BA0 BAPOH BDATZ BGNMA BSONS CAG COF CS3 CSCUP DDRTE DNIVK DPUIP DU5 EBLON EBS EIOEI EJD ESBYG FEDTE FERAY FFXSO FIGPU FINBP FNLPD FRRFC FSGXE FWDCC GGCAI GGRSB GJIRD GNWQR GQ6 GQ7 H13 HF~ HG6 HLICF HMJXF HRMNR HVGLF HZ~ IJ- IKXTQ ITM IWAJR IXC I~X I~Z J-C JBSCW JZLTJ KOV LLZTM M4Y MA- N2Q NB0 NPVJJ NQJWS NU0 O9- O93 O9J P2P P9R PF0 PT4 QOS R89 R9I RIG RNS ROL RSV S16 S1Z S27 S3B SAP SDH SHX SISQX SJYHP SMT SNE SNPRN SNX SOHCF SOJ SPISZ SRMVM SSLCW STPWE SZN T13 TSG TUC UG4 UOJIU UTJUX UZXMN VC2 VFIZW W23 W48 WH7 WK8 XU3 YLTOR YNT ZMTXR ~A9 AAPKM ABDBE ABFSG ABRTQ ACSTC AEZWR AFDZB AFHIU AFOHR AHPBZ AHWEU AIXLP ATHPR |
| ID | FETCH-LOGICAL-p141X-5ca31735f9f404ad2ef27dea9c9e1379405fb5cfcebbd8b371f29116fed652d93 |
| IEDL.DBID | AGYKE |
| ISSN | 0081-5438 |
| IngestDate | Thu Sep 25 00:56:24 EDT 2025 Fri Feb 21 02:38:22 EST 2025 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | Suppl 1 |
| Keywords | projective line vector bundle arithmetic surface torsion |
| Language | English |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-p141X-5ca31735f9f404ad2ef27dea9c9e1379405fb5cfcebbd8b371f29116fed652d93 |
| Notes | ObjectType-Article-1 ObjectType-Feature-2 SourceType-Conference Papers & Proceedings-1 content type line 22 |
| PQID | 3095150322 |
| PQPubID | 2044165 |
| ParticipantIDs | proquest_journals_3095150322 springer_journals_10_1134_S008154382403012X |
| PublicationCentury | 2000 |
| PublicationDate | 20240800 |
| PublicationDateYYYYMMDD | 2024-08-01 |
| PublicationDate_xml | – month: 8 year: 2024 text: 20240800 |
| PublicationDecade | 2020 |
| PublicationPlace | Moscow |
| PublicationPlace_xml | – name: Moscow – name: Heidelberg |
| PublicationTitle | Proceedings of the Steklov Institute of Mathematics |
| PublicationTitleAbbrev | Proc. Steklov Inst. Math |
| PublicationYear | 2024 |
| Publisher | Pleiades Publishing Springer Nature B.V |
| Publisher_xml | – name: Pleiades Publishing – name: Springer Nature B.V |
| References | Roberts (CR1) 1972; 24 Ranicki (CR9) 1985 Smirnov (CR4) 2015 Reidemeister (CR6) 1935; 11 Hanna (CR3) 1978; 52 Polyakov (CR10) 2022; 511 CR7 Horrocks (CR2) 1964; s3-14 Milnor (CR8) 1966; 72 Smirnov (CR5) 2018; 232 Cartan, Eilenberg (CR11) 1956 |
| References_xml | – year: 2015 ident: CR4 article-title: On filtrations of vector bundles over $$P^{1}_{\mathbb{Z}}$$ publication-title: Arithmetic and Geometry – volume: 24 start-page: 149 issue: 1 year: 1972 end-page: 154 ident: CR1 article-title: Indecomposable vector bundles on the projective line publication-title: Canad. J. Math. – volume: s3-14 start-page: 714 issue: 4 year: 1964 end-page: 718 ident: CR2 article-title: Projective modules over an extension of a local ring publication-title: Proc. London Math. Soc. – year: 1956 ident: CR11 publication-title: Homological Algebra – ident: CR7 – year: 1985 ident: CR9 article-title: The algebraic theory of torsion I. Foundations publication-title: Algebraic and Geometric Topology – volume: 511 start-page: 137 year: 2022 end-page: 160 ident: CR10 article-title: Finiteness of the number of classes of vector bundles on $$P^{1}_{\mathbb{Z}}$$ with jumps of height $$2$$ publication-title: Zap. Nauch. Sem. POMI – volume: 52 start-page: 322 issue: 2 year: 1978 end-page: 327 ident: CR3 article-title: Subbundles of vector bundles on the projective line publication-title: J. Algebra – volume: 232 start-page: 721 year: 2018 end-page: 731 ident: CR5 article-title: Vector bundles on $$P^{1}_{\mathbb{Z}}$$ with simple jumps publication-title: J. Math. Sci. – volume: 11 start-page: 102 year: 1935 end-page: 109 ident: CR6 article-title: Homotopieringe und Linsenräume publication-title: Abh. Math. Sem. Hamburg – volume: 72 start-page: 358 issue: 3 year: 1966 end-page: 426 ident: CR8 article-title: Whitehead torsion publication-title: Bull. Amer. Math. Soc. |
| SSID | ssj0045749 |
| Score | 2.2973418 |
| Snippet | We consider vector bundles of rank
2
with trivial generic fiber on the projective line over
ℤ
. For such bundles, a new invariant is constructed — the... We consider vector bundles of rank 2 with trivial generic fiber on the projective line over ℤ. For such bundles, a new invariant is constructed — the... |
| SourceID | proquest springer |
| SourceType | Aggregation Database Publisher |
| StartPage | S155 |
| SubjectTerms | Invariants Mathematics Mathematics and Statistics Topology |
| Title | Reidemeister Torsion for Vector Bundles on subscriptsuperscriptℙ1ℤ |
| URI | https://link.springer.com/article/10.1134/S008154382403012X https://www.proquest.com/docview/3095150322 |
| Volume | 325 |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVLSH databaseName: SpringerLink Journals customDbUrl: mediaType: online eissn: 1531-8605 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0045749 issn: 0081-5438 databaseCode: AFBBN dateStart: 20060101 isFulltext: true providerName: Library Specific Holdings – providerCode: PRVAVX databaseName: SpringerLINK - Czech Republic Consortium customDbUrl: eissn: 1531-8605 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0045749 issn: 0081-5438 databaseCode: AGYKE dateStart: 20060101 isFulltext: true titleUrlDefault: http://link.springer.com providerName: Springer Nature |
| link | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV27TsMwFLWgXWDgjSiUKgMbShW_4nhsUUMFagfUojBVcWwviLZqmoW5P8HOl_VLsPMAUVg6RIryknVzfe65vg8DcKOQDHxPM1f4lLuEedzlXBlnRVAqYMDNYQuFB0O_PyYPEY3KOu60ynavQpI5Uhf7jpC8ptfYexzYDnIGVqNdUKfWP6mBeuf-5bFXATChrGS9AXTtC2Uw89-P_CKWG7HQ3MSEh2BUDa7ILHltZ0vRTt43-jZuOfojcFBSTqdT6Mgx2FHTE7A_-O7Xmp6C8Mm2u3pT9p8vnNEsX0NzDJ91nvNVfaeb2WYMqWOuphXUpNnccMf8dL36gOvV5xkYh73RXd8td1hw55DAyKVJbPgDpppr4pFYIqURkyrmCVcQm6nqUS1oohMlhAwEZlAjg46-VtKnSHJ8DmrT2VRdAAcL5EnIEl8yY_CI4lIGOEYohoIZnylugGYl6Ek5TdIJtgSPegZUGuC2ktvP7dxBwWTyR3iXWz19BfaQoSJF2l4T1JaLTF0bKrEULaM6Ybc7bJUq9AXqJ8NG |
| linkProvider | Springer Nature |
| linkToHtml | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV27TsMwFL2CdgAG3ohCgQxsKFX8SuKxoJZCHwNKUZiqOHYWRFs1zcLcn2Dny_olOC8QhaVDJCtxIuvGPvdc34cBrhWWrm1Fjilsxk3qWNzkXGljRTAmkMv1lSYK9wd2Z0gffeYXedxxGe1euiQzpM7PHaFZTq_W98RNK8hpWPU3oUqR69IKVJv3L91WCcCUOQXrdZGZvlA4M__9yC9iueILzVRMew-8cnB5ZMlrI5mLRvi-UrdxzdHvw25BOY1mPkcOYEOND2Gn_12vNT6C9lNa7upNpf98ZniTbA_N0HzWeM529Y3bJC3GEBv6blxCTZxMNXfMmsvFB1ouPo9h2G55dx2zOGHBnCKKfJOFgeYPhEU8ohYNJFYRdqQKeMgVInqpWiwSLIxCJYR0BXFQhDU62pGSNsOSkxOojCdjdQoGEdiSyAlt6WiFRxWX0iUBxgESjraZghrUS0GPimUSj0hK8JilQaUGN6Xcfh5nBgqhoz_CO1ur9xVsdbx-b9R7GHTPYRtrWpKH8NWhMp8l6kLTirm4LKbRFxz-xLs |
| linkToPdf | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV07T8MwED5BKyEYeCMKBTKwoZT4lcRjgZZCaYVQi8pU4theEKUi7cLcP8HOL-svwU4TEI8FMUSKnIeS0_nuO_vuO4BDhWXoezpwhc-4SwOPu5wrE6wIxgQKuTlsoXCr7Te69LLHelmf0yTPds-3JGc1DZalaTA6Hkqd9SChaX2v8f0ktGxyxsT25qFI7VABitXzu2YtN8aUBRkCDpFrH8g2Nn99yReQ-W1fNHU39RW4zz90lmXyUBmPRCV--cbh-I8_WYXlDIo61ZnurMGcGqzDUuuDxzXZgPqNpcF6VFYXnp3OU7q25hic69ymq_3OydiSNCSOGU1yE5SMhwZTpqfTySuaTt42oVuvdU4bbtZ5wR0iinouiyODKwjTXFOPRhIrjQOpIh5zhYiZwh7TgsU6VkLIUJAAaWyspq-V9BmWnGxBYfA0UNvgEIE9iYLYl4FxhFRxKUMSYRwhEZhYKipBORd6P5s-SZ9Y4Mc8Y2xKcJTL8PNyGrgQ2v8hvJ0_3X0AC9dn9f7VRbu5C4vYoJVZZl8ZCqPnsdozaGMk9jONegcbGM2W |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=proceeding&rft.title=Proceedings+of+the+Steklov+Institute+of+Mathematics&rft.atitle=Reidemeister+Torsion+for+Vector+Bundles+on+subscriptsuperscript%E2%84%991%E2%84%A4&rft.au=Polyakov%2C+V+M&rft.date=2024-08-01&rft.pub=Springer+Nature+B.V&rft.issn=0081-5438&rft.eissn=1531-8605&rft.volume=325&rft.issue=1&rft.spage=S155&rft.epage=S167&rft_id=info:doi/10.1134%2FS008154382403012X&rft.externalDBID=NO_FULL_TEXT |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0081-5438&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0081-5438&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0081-5438&client=summon |